There are various known devices for applying medicament to the eye. A typical eye-drop container has a flexible vial and a nozzle for releasing drops of medicament into the eye by squeezing the vial. If the user squeezes too hard, too much medicament can be released and, as a result, drip down the user's cheek. There is typically no means provided for accurately controlling the volume of each dose of medicament released into the eye. Also, if the tip of the nozzle becomes contaminated, there is typically no means provided for preventing the contaminated tip from contaminating the medicament within the vial, and thus possibly causing infection by use of the contaminated medicament. Because the medicament in such devices is typically exposed to air upon opening the nozzle, it is not desirable to use such devices to administer preservative-free formulations of medicament, other than by providing single-dose containers. The single-dose containers, however, are typically relatively expensive to package, particularly when used for prescription medications.